# Properties

 Label 9.9.b Level $9$ Weight $9$ Character orbit 9.b Rep. character $\chi_{9}(8,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $1$ Sturm bound $9$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 9.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$9$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(9, [\chi])$$.

Total New Old
Modular forms 10 2 8
Cusp forms 6 2 4
Eisenstein series 4 0 4

## Trace form

 $$2 q + 476 q^{4} + 3304 q^{7} + O(q^{10})$$ $$2 q + 476 q^{4} + 3304 q^{7} - 8388 q^{10} - 52544 q^{13} + 104072 q^{16} + 93280 q^{19} + 181296 q^{22} - 1173154 q^{25} + 786352 q^{28} - 392888 q^{31} - 128844 q^{34} + 5638828 q^{37} - 4143672 q^{40} - 4426928 q^{43} + 2783952 q^{46} - 6071394 q^{49} - 12505472 q^{52} + 42241968 q^{55} + 5218308 q^{58} - 34810604 q^{61} + 20216432 q^{64} - 28645328 q^{67} - 13856976 q^{70} - 17813984 q^{73} + 22200640 q^{76} + 65517688 q^{79} + 5994252 q^{82} - 30020652 q^{85} + 89560224 q^{88} - 86802688 q^{91} - 13856112 q^{94} - 48903488 q^{97} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(9, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
9.9.b.a $2$ $3.666$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$3304$$ $$q+\beta q^{2}+238q^{4}+233\beta q^{5}+1652q^{7}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(9, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(9, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(3, [\chi])$$$$^{\oplus 2}$$